My first instinct was a sorted array with prefix sums, which gets you sample() in O(log n) via binary search but blows up on insert and delete.
Start by clarifying requirements and constraints, then propose a balanced binary search tree (e.g., an order-statistic tree) where each node stores the total weight of its subtree. For sampling, generate a random number between 0 and the root's total weight, then traverse down using subtree weight sums to find the corresponding item in O(log n).
Pro tip: Mention that this is essentially a Fenwick tree (Binary Indexed Tree) over a dynamic set, but since insert/delete require shifting, a balanced BST with subtree weight sums is more suitable for O(log n) operations. Also, discuss how to handle updates to weights.
Ask about the expected number of operations, whether weights can change, and if the data structure needs to be thread-safe. Confirm that O(log n) per operation is required for all three operations.
Propose a balanced binary search tree (e.g., AVL or Red-Black) where each node stores an item, its weight, and the sum of weights in its subtree. This allows efficient updates and weighted sampling.
For insert, add a new node and update subtree weight sums along the path to the root. For delete, remove the node and update sums similarly. Both operations take O(log n) time.
Generate a random integer between 1 and the total weight (stored at the root). Traverse from the root: at each node, compare the random value with the left subtree's weight sum; if smaller, go left; if larger, subtract and go right; if within the node's own weight, return the node's item.
Explain that all operations are O(log n) due to tree height. Discuss alternatives like Fenwick trees (O(log n) but with O(n) insert/delete due to shifting) and segment trees (similar but may require pre-allocation).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.